Coalition Structure Generation and CS-core: Results on
the Tractability Frontier for games represented by MC-nets
Julien Lesca
Patrice Perny
Makoto Yokoo
Université Paris-Dauphine
Place du Mal de Lattre de
Tassigny, Paris, France
CNRS, LAMSADE UMR 7243
Sorbonne Universités, UPMC
CNRS, LIP6 UMR 7606
4 Place Jussieu, Paris, France
Kyushu University
Motooka 744, Fukuoka, Japan
[email protected]
[email protected]
ABSTRACT
of value produced by the set of coalitions). This problem,
which is called coalition structure generation (CSG), has
been widely investigated in the literature (e.g. [30, 33, 24],
see [10] and rahwan15 for a survey on that topic). The CSG
problem is highly combinatorial due to the high number1 of
possible coalition structures. However, for some subclasses
of games (e.g., super-additive games) the CSG problem is
easy to solve since the grand coalition is known to be optimal. The CSG problem for games represented by MC-nets
has been investigated (e.g. [33, 21, 32]) and proven to be
NP-hard in [25]. Various polynomial-time solvable classes of
games have also been studied for it [3, 34, 6, 4]. It is worth
continuing the study of easy and hard cases for other classes
of MC-nets to further specify the frontier of tractability for
this problem. One contribution of this paper is to study
games represented by a specific class of MC-nets and identify both hard and easy cases for the CSG problem.
Once coalitions are formed we need to define how the welfare (or value) obtained by each one is shared among participants. In an environment with money, a frequent assumption consists of considering transferable utilities which
supposes that the overall gain or welfare obtained by a coalition can be freely divided among agents (for example using
money transfers). In that case, we need to define how important each agent is to the overall cooperation, and what
payoff she can reasonably expect. Knowing that agents may
collude to obtain more money, we have to impose some stability properties to reach an equilibrium. One simple way
of defining this stability is to consider the core concept [17].
Informally, a solution belongs to the core if no subset of
agents has an incentive to deviate. The concept of the core
has been generalized [2], under the name of CS-core, to cases
where agents are not constrained to form the grand coalition. Unfortunately there are situations where this stability
cannot be attained because the game’s CS-core is empty. On
the other hand, many families of games have a non-empty
CS-core, e.g. balanced games [29, 20], and looking for a solution in the CS-core makes sense to ensure stability. The
complexity of computing solutions belonging to the core has
been widely studied in AI (e.g. [12, 15, 1, 18, 23]) and polynomial cases have been identified (e.g. [7, 8, 9, 5]). However,
most of these works focused on the classical notion of core
and very few considered the more general definition of CScore. Testing the emptiness of the CS-core is ∆P
2 -complete
[19]. In the second part of this paper, we define a new class of
The coalition structure generation (CSG) problem consists
in partitioning a group of agents into coalitions to maximize the sum of their values. We consider here the case of
coalitional games whose characteristic function is compactly
represented by a set of weighted conjunctive formulae (an
MC-net). In this context the CSG problem is known to be
computationally hard in general.
In this paper, we first study some key parameters of MCnets that complicate solving make the CSG problem. Then
we consider a specific class of MC-nets, called bipolar MCnets, and prove that the CSG problem is polynomial for
this class. Finally, we show that the CS-core of a game
represented by a bipolar MC-net is never empty, and that
an imputation belonging to the CS-core can be computed in
polynomial time.
1.
[email protected]
INTRODUCTION
Cooperative game theory develops a set of mathematic
tools for modeling various situations involving rational agents
and analyzing how groups of agents may adopt cooperative
behaviors. These tools provide important foundations for
the design and analysis of multiagent systems. One important issue in cooperative games is to analyze coalition
structures, i.e. partitions of the set of agents into subsets of
agents who perform coordinate actions.
Formally a cooperative game is characterized by a set
function that provides the value of every possible coalition
of agents. Nevertheless the explicit representation of this
set function becomes exponentially large as the number of
agents increases. An important issue is therefore to describe
the game as succinctly as possible. Several schemes have
been proposed for the compact representation of games see
e.g. [14, 12, 1]. Among them, MC-nets describing games
using a set of weighted logic formulae [22] are presented as
a simple and intuitive language. They have received much
attention in the last few years [25, 16, 23].
For a given game, an important concern is to find the optimal organization of cooperations among agents. One way
of addressing this problem is to look for a coalition structure that maximizes the social welfare (i.e., the total amount
Appears in: Proceedings of the 16th International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2017), S. Das, E. Durfee, K. Larson, M. Winikoff
(eds.), May 8–12, 2017, São Paulo, Brazil.
c 2017, International Foundation for Autonomous Agents and
Copyright 1
n
The number of possible coalition structures is in Ω(n 2 )
[27], where n is the number of agents
Multiagent Systems (www.ifaamas.org). All rights reserved.
308
MC-nets and prove that the games representable in it have a
non-empty CS-core, and computing a solution that belongs
to the CS-core can be performed in polynomial time. Some
polynomial cases for CS-cores have been recently identified
[11], under some restrictions about possible coalition structures. The polynomial cases presented in this paper do not
require such restrictions.
2.
no rule containing only negative literals is used. Furthermore, analogously to previous work [25], we focus here on
basic MC-nets with positive weights, i.e. any weight vi in a
rule of P must be strictly positive. Such a restriction on the
representation does not harm the expressivity because it is
well known that any game with positive values can be represented by a basic MC-net with positive weights (e.g. [31]).
Even under such restrictions on MC-nets, the following result (Theorem 2 of [25]) shows that CSG is computationally
hard to solve.
PRELIMINARIES
For any i ∈ N, we denote by [i] set {1, . . . , i} and consider
a set of n agents N = [n] who are willing to cooperate to
improve their incomes. The income for any subset of agents
is defined by a set function v : 2N → R+ such that v(∅) = 0,
which is called a game. For any S ⊆ N value v(S) represents
the total amount of money that agents of S can share if they
cooperate. In the following, let us call coalition any subset
of cooperating agents.
A coalition structure π = {S1 , . . . , Sk } is a partition of N
into coalitions. Let ΠN be the set of all possible coalition
structures on N . We assume that agents are free to form
any possible coalition structure. The value of a coalition
structure is defined additively as follows.
Theorem 1. Finding an optimal coalition structure is
strongly2 NP-hard. Moreover, unless P = NP, there exists
no polynomial-time O(|P |1− )-approximation algorithm for
any > 0.
In the following subsections, we describe a graphical representation of the CSG problem for basic MC-nets with positive weights and study the impact of some parameters of this
representation on the problem’s complexity.
3.1
Definition 1.PThe value of a coalition structure π is defined by v(π) = S∈π v(S).
A coalition structure π ∗ ∈ ΠN is said to be optimal whenever
v(π ∗ ) = max{v(π) : π ∈ ΠN }. The CSG problem consists
in finding such an optimal coalition structure π ∗ .
Describing a game by its characteristic form, i.e. the set
of values v(A) for all A in N , may become unfeasible as n
growths, since 2n values are needed. In this paper we assume
that games are represented by basic MC-nets [22]. A basic
MC-net is a finite set of rules {φi : (Pi , Ni ) → vi }i∈K , where
Pi ⊆ N , Ni ⊂ N , Pi ∩ Ni = ∅, vi ∈ R \ {0}, and K is the set
of indices for these rules. A rule φi is said to be applicable to
coalition S if Pi ⊆ S and Ni ∩S = ∅, i.e. where S contains all
the agents in Pi and no agent in Ni . By extension, φi is said
to be applicable to a coalition structure π if it is applicable
to a coalition S belonging to π. The set of indices of rules
applicable to S is denoted as A(S). Hence, the value of any
coalition S is given by the sum ofPthe weights of all rules
applicable to S. Formally, v(S) = i∈A(S) vi .
• set of vertices V is P (vertex i represents rule φi );
• green edge {i, j} in E is associated to any pair of rules
(i, j) ∈ P 2 such that Pi ∩ Pj 6= ∅;
• red edge {i, j} in E is associated to any pair of rules
(i, j) ∈ P 2 such that Pi ∩ Nj 6= ∅.
Example 1. Consider MC-net {φ1 : ({1, 3}, ∅) → 3, φ2 :
({1, 2}, ∅) → 3, φ3 : ({1}, {2}) → 2}. Let S = {1, 2, 3}.
We have A(S) = {1, 2}, corresponding to rules φ1 and φ2 .
Hence v(S) = 3 + 3 = 6. It can easily be checked that {S}
is the optimal coalition structure.
An instance of a CSG-graph is given in Fig. 1 in the case of
Example 1 (red edges are represented as dotted lines).
One can interpret any pair (Pi , Ni ) as a conjunctive formula including the positive literals of Pi and the negative
literals of Ni . Hence, basic MC-nets form a subclass of more
general MC-nets that may include other forms of weighted
formulae [22]. Yet, they are still fully expressive for representing games.
3.
CSG-graph of an MC-net
Each rule φi in P supports cooperation between agents in
the sense that the agents in Pi have to belong to the same
coalition to obtain reward vi . A set of rules R ⊆ P is said
to be feasible if there exists a coalition structure π where
each rule in R is applicable to π. As highlighted by [25], the
CSG problem can be reformulated to find a feasible subset
of rules in P that maximizes the sum of their weights.
As an illustration, let us return to Example 1. The feasible sets of rules are {φ1 , φ2 }, {φ1 , φ3 } and any subset of
them, including the empty set. For example, both φ1 and φ3
are applicable to coalition structure π = {{1, 3}, {2}} which
makes {φ1 , φ3 } a feasible set. Here the optimal feasible set is
{φ1 , φ2 } with a value 6 corresponding to the grand coalition.
The feasibility of a set of rules results from the interactions
between its rules. These interactions are complex but can
be modeled by a colored multigraph G = (V, E), called a
CSG-graph3 , defined as follows:
Figure 1: CSG-graph of Example 1
COMPUTATIONAL ISSUES IN CSG
Let Gg denote the subgraph of G that is restricted to the
green edges, and let Gr be the subgraph of G that is restricted to the red edges. Furthermore, let Gg [R] denote the
Given an MC net, we distinguish two types of rules. Let
P = {i ∈ K : Pi 6= ∅}, and let P̄ = {i ∈ K : Pi = ∅} be
its complement. By definition, the rules in P are applicable
at most to one coalition within a given coalition structure,
whereas the rules in P̄ may be applicable to several coalitions. In this section, we assume that P̄ is empty, i.e. where
2
The strong NP-hardness was not previously stated but this
result is implied by the proof of Theorem 2 in [25].
3
Note that a CSG-graph is a simplified version of the graphical representation described in [25].
309
j ∈ V, {i, j} ∈ H, i < j}, Ni = {aij : i ∈ V, {i, j} ∈ H, i >
j} and vi = w(i). By construction, optimal solution A of
WVMC has value w(V) − v(π ∗ ), where π ∗ is the optimal
coalition structure of the game defined by the MC-net.
vertex-induced subgraph of Gg restricted to the vertices of
R ⊆ P . For given coalition structure π, if two rules φi and
φj are connected by a green edge {i, j}, then they are applicable to the same coalition S ∈ π such that Pi ∪ Pj ⊆ S.
On the other hand, if i and j are connected by a red edge
then rules φi and φj are not applicable to the same coalition
of π due to condition Pi ∩ Nj 6= ∅. This implies that for
any coalition structure π and any pair of rules {i, j} that
are connected by a red edge, if i and j are also connected by
a path containing only green edges, at least one rule visited
by this path is not applicable to π. Hence, as shown4 in [25]
(Theorem 1), set of rules R is feasible if and only if no connected component of Gg [R] contains a pair {i, j} connected
by a red edge in G. Identifying a feasible set of rule with
maximum weight (i.e. for solving the CSG problem) can be
achieved by solving the weighted vertex multicut problem:
This proposition provides evidence that, for games represented by basic MC-nets with positive weights, the complexity of the CSG problem is related to the number of red edges
in CSG-graphs. Unfortunately, even if the number of red
edges of a CSG-graph is a fixed parameter, there is no hope
to find a polynomial algorithm. In other words, the CSG
problem does not belong to the parameterized complexity
class XP, when the fixed parameter under consideration is
the number of red edges in the CSG-graph.
Another complexity issue is the approximability of the
CSG problem for games represented by basic MC-nets with
positive weights. Actually, a |P |-approximation can easily
be obtained by choosing the rule φi of P with the highest
weight vi , and by constructing a coalition structure where
the agents of Pi form a coalition and the other agents form
singleton coalitions. Theorem 1 shows that this is essentially
the best approximation ratio achievable, up to a multiplicative constant factor. Now the question is whether we can
achieve a better approximation ratio by fixing some parameters of the problem. The following proposition shows that
a small number of red edges in the CSG-graph enables a
better approximation ratio than O(|P |).
Weighted Vertex Multicut (WVMC)
Input: Undirected graph G = (V, E), collection H ⊆ V 2 of
pairs of vertices, and weights w(x) ∈ R≥0 , for x ∈ V.
Question: Find A ⊆ V whose
P removal separates each pair
in H and such that w(A) = x∈A w(x) is minimum.
To solve the CSG problem, we just have to solve the above
WVMC problem on graph Gg with H defined as the set
of red edges and w(i) = vi for any vertex i of Gg . The
optimal set of rules R we are looking for is given by the
complement of A in V . By minimizing the weight of A we
indeed maximize the weight of its complement.
Let us illustrate this point on the graph of Example 1 (Fig.
1). Pair {2, 3} is connected with a red edge but also by two
green paths (one direct and one including vertex 1). Hence
to cancel both paths, we need to remove either vertex 2 or
3. Here the optimal cut is A = {3} because w(2) > w(3)
and the optimal feasible set is R = {1, 2}.
3.2
Proposition 2. Whenever the CSG-graph contains no
more than k red edges, it is possible to compute in polynomial
time a k-approximation of the optimal coalition structure.
Proof. Let C1 , C2 , . . . , Ct be the connected components
of Gr that contain at least two vertices. Any vertex of these
connected components has a degree greater than or equal to
one. For any i ∈ {1, ..., t}, and j ∈ {1, . . . , k}, we define Cij
as follows. If Ci contains at least j vertices of degree greater
than or equal to 2 then Cij is a singleton containing the j th
vertex of a degree greater than or equal to 2 in Ci , where
the order is taken arbitrarily. Note that at most k vertices
can have a degree greater than or equal to 2 in a graph
containing no more than k edges. If Ci contains exactly 2
vertices then Ci1 is a singleton that contains one of these
vertices taken arbitrarily. For other values of i and j, Cij is
an empty set.
Consider the following k sets of rules:
S
• Rj = ti=1 Cij , for j = 1, . . . , k − 1
S
• Rk = P \ k−1
j=1 Rj
Complexity issues and CSG-graph
First, note that the WVMC problem can be solved independently in the different connected components of G. This
implies that the CSG-problem can also be solved independently in the different connected components of Gg . In the
following, we assume that a CSG-graph is such that Gg is
composed by a unique connected component.
Even though the WVMC problem is known to be NPhard [13] for |H| ≥ 3, it is polynomial for |H| ≤ 1 (e.g.,
for |H| = 1 it boils down to a mincut problem). The latter
property suggests a polynomial algorithm to solve instances
when the CSG-graph includes at most one red edge. It is
indeed sufficient to find a vertex mincut that separates the
two vertices linked by a red edge in G (vertex mincut is a
variant of min cut where cuts are made of vertices instead
of edges). On the other hand we have:
For any j ∈ [k − 1], Rj contains at most one rule per connected component of Gr . Therefore, these sets are feasible
since they cannot contain a pair of rules connected by a red
edge. Furthermore, if Rk does not contain a vertex of degree higher than 1 in Gr then it cannot contain two rules
connected by a red edge, except if those two rules are the
two members of a connected component of Gr that contains
exactly two vertices. But in the later case, we know that one
rule is part of R1 and the other is part of Rk . On the other
hand, if Rk contains a vertex of degree more than two then
there is only one connected component containing at least
two vertices in Gr i.e., t = 1. This connected component
C1 is a cycle containing exactly k vertices, and Rk contains
only one of them. Therefore Rk is also feasible.
Proposition 1. The CSG problem is NP-hard even if the
CSG-graph includes at most three red edges.
Proof. The proof is based on a reduction from WVMC
with parameters G = (V, E) and H. We introduce one agent
ai per vertex i ∈ V and one agent aij per pair {i, j} ∈
H. Then, we define an MC-net with exactly one rule φi by
vertex i ∈ V, with Pi = {ai } ∪ {aj : j ∈ V, {i, j} ∈ E} ∪ {aij :
4
Note that the statement of Theorem 1 in [25] is slightly
different due to the difference between the two graphical
representations. However, the implication of this theorem
on CSG-graph representation is straightforward.
310
Among
P such k feasible sets, let Rj ∗ be a set that maximizes i∈Rj vi . It is easy to verify that the value of Rj ∗ is
P
at least equal to i∈P vi /k. Hence, the coalition structure
associated with Rj ∗ has a value greater than or equal to 1/k
times the value of the optimal coalition structure.
subsets of S in a coalition structure. Therefore whenever
v is positive, it is always beneficial to split a coalition into
subcoalitions to collect several times the value v (once for
each subset of S).
Note that the proof of Proposition 2 remains valid if k is
the size of the largest connected component of Gr . Therefore
the following result also holds:
Similar to the case where P̄ is empty, we solve the CSG
problem by computing the feasible subset of rules in P with
the highest weight. But now, the weight of feasible set of
rules R not only depends on the rules that are part of R,
but also on the rules of P̄ . To compute the weight provided
by those rules to a feasible subset of rules R, we need first
to define coalition structure π where all rules of R are applicable. Then we can compute v̂(π), i.e. the weight of π
provided by the rules of P̄ .
For each feasible subset of rules R, there exists a whole
set Π(R) of coalition structures for which all the rules of R
are applicable. For example, for feasible subset of rules {φ2 }
of Example 2, both π1 = {{1, 2}, {3}} and π2 = {{1, 2, 3}}
belong to Π({φ2 }). Among them, π1 seems most promising
since v̂(π1 ) = 2 > v̂(π2 ) = 0. Within Π(R), let π(R) denote the most refined coalition structure where the notion
of refinement is defined as follows. Coalition structure π 0
is a refinement of π when any coalition of π is the union
of the coalitions of π 0 . By construction, π(R) is a coalition
structure maximizing v̂ over Π(R). Therefore π(R) is a good
candidate to represent a feasible set of rules R. Based on
this observation, we define the weight of a subset of rules R
as follows:
P
i∈R vi + v̂(π(R)) if R feasible
w(R) =
(1)
0
otherwise
4.1
Corollary 1. Whenever the size of the largest connected
component of the red edges in the CS-graph equals k, it is
possible to compute in polynomial time a k-approximation of
the optimal coalition structure.
To summarize this section, we provided evidence that the
number of red edges in the CSG-graph contributes to the
complexity of the CSG problem.
4.
TOWARDS MORE GENERAL RULES
In this section we consider a more general case where P̄
is nonempty and propose an algorithm to solve the CSG
problem for that specific case. In P̄ all rules are assumed
to have positive weights, as in P . Note that, by definition,
all the rules of P̄ apply to the empty set. However, since
the value of the empty coalition equals 0, we assume that
no rule of P̄ applies to the empty set. This assumption can
be made without loss of generality since the empty set does
not play a role in the CSG problem. We denote by v̂ the
restriction of
Pv to the rules of P̄ , i.e. where for any coalition
S, v̂(S) = i∈P̄ :S∩Ni =∅ vi , and for any coalition structure
P
π, v̂(π) = S∈P v̂(S).
Example 2. Modify Example 1 by adding rule φ4 : (∅, {1}) →
2. With this new set of rules, the value of the grand coalition N remains unchanged because φ4 does not apply to this
coalition. On the other hand, the value of coalition structure
π = {{1, 3}, {2}} becomes 3 + 2 + 2. It is easy to check that
π is now the optimal coalition structure for this new set of
rules.
New approach to solve CSG problem
Note that for a given feasible set of rules R, w(R) may differ
from v applied to π(R). Indeed, some rules of P \ R may
apply to π(R). However, the following proposition shows
that the CSG problem can be solved by finding the feasible
subset of rules that maximizes w:
Proposition 3. If R∗ ⊆ P maximizes w, then π(R∗ ) is
an optimal coalition structure of game v.
Proof. Note that π(∅) is a coalition structure that includes one coalition per agent. Furthermore, P̄ includes at
least one rule i with strictly positive weight vi . This rule
applies to at least one singleton in π(∅) and implies that
v̂(π(∅)) ≥ vi > 0. Therefore w(π(R∗ )) > 0 and R∗ is necessarily feasible.
For any feasible set of rules R ⊆ P there exists at least
one feasible set of rules R0 ⊆ P such that R0 ⊇ R, π(R) =
π(R0 ) and w(R0 ) = v(π(R0 )). Set R0 contains all of the
formulae of P that are true for at least one coalition of π(R).
Since ∀R, R0 ⊆ P such that R ⊆ R0 and π(R) = π(R0 ) we
have w(R) ≤ w(R0 ), and then π(R∗ ) is the best coalition
structure in {π(R)|R ⊆ P, R feasible}.
Let us prove now that the optimal coalition structure
necessarily belongs to {π(R)|R ⊆ P, R feasible}. Let π ∈
ΠN \{π(R)|R ⊆ P, R feasible}. Let R ⊆ P be the maximal
subset of rules in P that apply to π. Note that R is feasible.
For any coalition S ∈ π, by definition of π(R), there exists a
subset of coalitions π 0 ⊆ π(R) such that π 0 forms a partition
of S. If we divide S according to π 0 , then the rules in R remain applicable to π(R). Let C be the subset of rules of P̄
that apply to S. For any S 0 ∈ π 0 , we have S 0 ⊆ S and therefore all the formulae in C are also applicable to S 0 (since for
As highlighted in [25], a rule of P̄ can be simulated by
a polynomial number of rules from P . For example, rule
(∅, N \ [k]) → v can be simulated using k rules φ1 : ({1}, N \
[1]) → v, φ2 : ({2}, N \ [2]) → v, . . . , φk : ({k}, N \ [k]) →
v. However, these rules introduce multiple red edges in the
CSG-graph, and we know from Section 3.2 that they are a
source of complexity. In this example, there is a red edge
between any pair of new rules {φi , φj }, and Gr contains a
clique of size k. Moreover, φ1 contains almost all of the
agents in N1 . Such a rule will be connected in Gr to almost
all the rules and create a large connected component in Gr .
This section propose another way of dealing with the rules
of P̄ .
Let us extend our discussion about the expressivity of the
rules belonging to P̄ . For a given set of agents S, rule
(∅, N \ S) → v assigns weight v to any subset of S. The
particularity of this rule, compared to the rules of P , is
that it can be applied to several coalitions of a coalition
structure. For example, rule φ4 of Example 2 can be applied to coalitions {1} and {3}. Hence φ4 can be applied
twice to coalition structure {{1}, {2}, {3}}. Actually, a rule
(∅, N \ S) → v can be applied as many times as there are
311
any i ∈ P̄ , S ∩ Ni = ∅ ⇒ S 0 ∩ Ni = ∅). Since the weights
are strictly positive, then applying this subdivision to any
coalition of π implies v(π) ≤ v(π(R)). So an optimal coalition structure necessarily belongs to {π(R)|R ⊆ P }, which
concludes the proof.
4.2
any i < k. Because i and j are not part of the same coalition
in π 0 , there must be at least one rule of this sequence that
does not apply to π 0 . This leads to a contradiction.
To summarize, the computation of w(R) for any R ⊆ P
can be performed in polynomial time. One possible way of
solving the CSG problem i.e., finding the feasible subset of
rules that maximizes w (see Proposition 3), is to use a bruteforce algorithm to test every subsets of rules of P . This is
an FPT algorithm when the fixed parameter is the size of
P . However, if |P | is large then such a brute-force algorithm
will be inefficient. In that case, it would be interesting to
investigate the properties of set function w. Once the set of
rules R∗ maximizing w is computed, the optimal coalition
structure can be derived in polynomial time from R∗ by
determining π(R∗ ).
Before discussing polynomial cases, note that the proofs of
Propositions 3 and 4 can be extended to any set function v̂
where splitting coalitions is always beneficial i.e., to any set
function such that v̂(S) ≤ v̂(π) for all S ⊆ N and π ∈ ΠS .
Hence the algorithm remains valid even if we enrich the class
of basic MC-nets with positive weights by allowing extra
rules of the following types5 :
Computing optimal coalition structure
To compute the value of w for a given subset of rules
R ⊆ P , we need first to define whether R is feasible and
then compute π(R). Deciding whether R is feasible can be
performed in polynomial time. We only have to inspect all of
the connected components of Gg [R] and check that none includes two vertices connected by a red edge in G. Moreover,
π(R) can be computed by regrouping agents by the rules of
R as described by the following polynomial algorithm.
Algorithm 1: Algorithm to compute π(R)
Data: Feasible set of rules R and CSG-graph G
Result: Coalition structure π ∈ ΠN
π ← {{}};
A ← N;
foreach S
connected component C of Gg [R] do
S ← i∈C Pi ;
π ← π ∪ {S};
A ← A \ S;
end
foreach i ∈ A do
π ← π ∪ {i};
end
return π;
• conjunction of positive
literals with negative weight
V
i.e., rules of type { j∈Pi xj } → vi , with vi < 0;
• disjunction of positive
literals with positive weight i.e.,
W
rules of type { j∈Pi xj } → vi , with vi > 0;
• disjunction of conjunction of negative literals with positive
W weight i.e. rules
V of type
{ N` ∈{N1 ,...,Nk } j∈N` ¬xj } → vi , with vi > 0.
The correctness of Algorithm 1 is established as follows.
Therefore this algorithm can be applied to a broad class of
MC-nets. However, since the polynomial case of Section 4.3
does not include these rules, they have been deliberately
omitted for simplicity and clarity.
Proposition 4. For any feasible set of rules R, the coalition structure returned by Algorithm 1 corresponds to π(R).
Proof. To establish the result we prove that all of the
rules of R are applicable to coalition structure π returned
by Algorithm 1, and no coalition of π can be split without
loosing this property. Let us first show that all rules of R
are applicable to π. By contradiction, we assume that there
exists a rule in R that is not applicable to π. Without loss
of generality, let φ1 be this rule. By construction of π, there
exists a coalition S in π such that P1 ⊆ S. If φ1 does not
apply to S then N1 ∩S 6= ∅. This implies by the construction
of π that there is a sequence of rules φ1 , φ2 , . . . φk belonging
to R such that Pi ∩ Pi+1 6= ∅ for any i < k, and N1 ∩ Pk 6= ∅.
But this sequence implies that there is a path of green edges
from φ1 to φk in Gg [R], and φ1 and φk are linked by a
red edge in G. This leads to a contradiction with R being
feasible.
Let us show now that no coalition of π can be split while
preserving the applicability of all the rules in R. Assume, by
contradiction, that there exists refinement π 0 of π in Π(R),
resulting from π by splitting at least one of its coalitions S.
Note that S cannot be a singleton. This implies that there
exists a subset
S of rules RS ⊆ R connected in Gg [R] and such
that S = i∈RS Pi . Let i and j be two agents belonging to
S who are not in the same coalition in π 0 . Let φ1 and φk be
two rules of RS such that i ∈ P1 and j ∈ Pk . Since φ1 and φk
are part of the same connected component of Gg [R] which
contains the rules of RS , there must be a sequence of rules
φ1 , φ2 , . . . φk belonging to RS such that Pi ∩ Pi+1 6= ∅ for
4.3
Polynomial cases for CSG problem
In general, it is impossible to compute the maximum value
of a set function without checking all of its values. However,
the problem is known to be solvable in polynomial time for
supermodular set functions, which are defined as follows:
Definition 2. Set function w over 2P is supermodular
whenever for every, A, B ⊆ P such that B ⊆ A, and for
every j ∈ P \ A, w(A ∪ {j}) − w(A) ≥ w(B ∪ {j}) − w(B).
If we were to identify a restriction on basic MC-nets, and
more precisely on the set of rules P , such that the set function described by (1) is supermodular, then the CSG problem would be solvable in polynomial time for this restricted
language. We know that using rules of P does not simplify
solving the CSG problem. Therefore according to Proposition 1, there is no hope of finding a polynomial case without
restricting the number of red edges of the CSG-graph to a
value lower than 3. The following proposition shows that
when the CSG-graph does not contain any red edges, set
function w is supermodular.
5
The rules are described by logic formulae over the set of
boolean variables {x1 , . . . , xn }. For each agent i, xi is true
if i is part of the coalition, and false otherwise. The weight
of a rule is attributed to a coalition if the logic formula is
true. This description is consistent with MC-nets [22]
312
Proposition 5. If P is feasible then set function w defined by (1) on 2P is supermodular.
define the team’s value or efficiency, we may add the individual values resulting from rules of type φi : ({i}, ∅) → vi
where vi is the intrinsic value of agent i. However, some subsets of agents have an inclination to work efficiently together
which will provide to their coalition an additional value. For
such subset of agents S, this can be modeled by inserting a
rule φS : (S, ∅) → vS , where vS corresponds to the added
value due to positive synergy.
On the other hand, every agent i works at a different maximal speed si , depending on her skill, and it may happen that
the group’s working speed is constrained by the speed of its
slowest member. Considering just this aspect, the value of
coalition S can be defined as mini∈S si . Such negative synergies can be represented by a linear number of rules of P̄ .
Consider indeed n agents such that s1 ≥ s2 ≥ . . . ≥ sn .
For i = 1, . . . , n, we insert in P̄ rule φi : (∅, {1, . . . , i}) →
si − si+1 , assuming sn+1 = 0. With such rules, it is easy to
check that the value of any coalition S induced by the rules
of P̄ is v̂(S) = mini∈S si .
Proof. Let A, B ⊆ P such that B ⊆ A and j ∈ P \
{A}. Since P is feasible,
any subset R ⊆ P is also feasible.
P
Therefore
w(R)
=
v
i∈R i +v̂(π(R))
P
P
P by (1). Now,
P observing
that i∈A∪{j} vi − i∈A vi = i∈B∪{j} vi − i∈B vi = vj
, we just need to show the following:
v̂(π(A ∪ {j})) − v̂(π(A)) ≥ v̂(π(B ∪ {j})) − v̂(π(B)) (2)
This property is proved in Lemma 1 of Section Appendix.
The following corollary summarizes the main implication
of Proposition 5 for the CSG problem:
Corollary 2. If the CSG-graph does not contain any red
edges, then the CSG problem can be solved in polynomial
time.
Proof. By definition, we know that if the CSG-graph
does not contain any red edges, then P is feasible. Furthermore, by Proposition 5, w is supermodular because P
is feasible. We can resort to a polynomial time algorithm
(e.g. [26]) to find R∗ ⊆ P that maximizes w. By Proposition 3, π(R∗ ) is an optimal coalition structure that can
be computed in polynomial time from R∗ (see Proposition
4).
Thus we have identified a polynomial subcase for the CSG
problem characterized by basic MC-nets with positive weights.
Since the NP-hardness of the CSG problem starts at the 3
red edges, the cases with 1 and 2 edges remain open. Note
that without the rules of P̄ , the resolution is trivial for the
case without red edges since feasible set P would always be
optimal. However, this is not the case whenever P̄ is not
empty.
5.
BIPOLAR MC-NETS
Finally, positive and negative synergies among agents can
be considered together by merging in the same bipolar MCnet the two sets of rules introduced above, to define the game
as the combination of the two phenomena.
More generally, analogously to the fact that rules of P
describe belief functions, the descriptive power of the rules
of P̄ can be analyzed through sum-min games6 defined as
follows. Game v̂ is a sum-min game if there exists integer
m ∈ N and
Pmm × n matrix s of non-negative values, such that
v̂(S) =
i=1 minj∈S sij , where sij denotes the value of s
at row i and column j. The illustrative example provided
in this section uses matrix s of size 1 × n. The following
proposition shows that the set of games representable by
bipolar MC-nets is essentially equivalent to the set of games
which are additive combinations of belief functions and summin games.
We introduce now a new subclass of basic MC-nets with
positive weights, called bipolar MC-nets, defined as follows.
Proposition 6. Game v̂ is sum-min if and only if it can
be described by a bipolar MC-net with no rule in P .
Definition 3. A bipolar MC-net is a basic MC-net characterized by a set of rules {φi : (Pi , Ni ) → vi }i∈K such that
Pi = ∅ or Ni = ∅, Pi ∪ Ni 6= ∅ and vi > 0, for all i ∈ K.
Proof. The description provided in Example 3 shows
how value minj∈S sij can be expressed through a set of rules
of P̄ . It remains for us to show that any rule φi : (∅, Ni ) → vi
defines a sum-min game. Let s = (s1 , . . . , sn ) be a vector
such that sj = vi for any j ∈ N \ Ni , and sj = 0 otherwise.
It is easy to check that minj∈S sj equals vi if S ∩ Ni = ∅,
and 0 otherwise. Therefore the game described by φi is a
sum-min game.
The rules of bipolar MC-net are therefore partitioned into
sets P and P̄ , where P contains all the preconditions of type
(Pi , ∅), and P̄ contains all the preconditions of type (∅, Nj ).
Note that the CSG-graph contains by definition no red edge
since Nj = ∅ for all j ∈ P . This implies by Corollary 2
that the CSG problem on a game represented by a bipolar
MC-net is solvable in polynomial time.
5.1
The proof of Proposition 6 also shows that the transformation from one representation to another cannot increase the
size of the description by a factor higher than n.
Expressivity of bipolar MC-nets
In a bipolar MC-net, the descriptive powers of the rules of
P and P̄ are antagonist. On one hand, a rule φi : (Pi , ∅) →
vi assigns a positive weight vi to any superset of Pi . This
descriptive power is well known in the literature of decision
theory under the name of belief functions [28]. On the other
hand, a rule φj : (∅, Nj ) → vj assigns a positive weight vj
to any subset of N \ Nj . This language is obviously not
fully expressive, but the following example illustrates the
expressivity of bipolar MC-nets.
5.2
Example 3. Consider a group of agents which must be
partitioned into teams, to collectively complete a project. To
6
The sum-min games introduced in this paper are defined
in a similar fashion as the max-game [9].
Computation of CS-core
Once coalitions are formed we need to share the payoff
among the participants of any coalition. This is formalized
by the notion of imputation:
Definition 4. An imputation is a pair hπ, xi with π ∈
ΠN and x = (x1 , . . . , xn ) ∈ Rn , and a vector of payoffs
such that for all i ∈ N , xiP≥ v({i}) and for all S ∈ π,
x(S) = v(S), where x(S) = i∈S xi .
313
Pn
P
xj = v(T ∪ {n}) + j∈[n−1]\T xj . We denote by s
the highest index in [n − 1] \ T . By induction hypothesis,
we know that there exists P
partition πs = {S1 , . . . , Sk } of
the agents in [s] such that sj=1 xj ≤ v(πs ). This implies
P
P
P
that j∈[n−1]\T xj ≤ kj=1 v(Sj ) − j∈T ∩[s] xj holds. We
denote by B = {b1 , . . . , bt } the subset
P of indices in [k] such
that Sbr ∩ T 6= ∅. We know that j∈T ∩Sb xj ≥ v(T ∩ Sbr )
r P
holds for any r ∈ [t]. This implies that
j∈T ∩[s] xj =
Pt
Pt P
v(T
∩
S
).
Altogether,
we have
x
≥
j
b
r
r=1
r=1
j∈T ∩Sbr
Pt
Pk
Pn
r=1 v(T ∩ Sbr ).
j=1 v(Sj ) −
j=1 xj ≤ v(T ∪ {n}) +
On the other hand, we know by Lemma 2 that v(Sbi ) +
Si
S
v( i−1
r=1 Sbr ∪ T ∪ {n}) ≤ v(T ∩ Sbi ) + v( r=1 Sbr ∪ T ∪ {n})
holds
obtain v(T ∪ {n}) +
St(see Appendix). We P
Pt for any i ∈ [t]
t
r=1 v(T ∩ Sbr ) by
r=1 v(Sbr ) ≤ v( r=1 Sbr ∪ T ∪ {n}) +
summing all these inequalities over i. Finally, by combining
this inequality with the
shown in the previous
P inequality S
xj ≤ v( tr=1 Sbr ∪ T ∪ {n}) +
paragraph, we obtain n
j=1
P
j∈[k]\B v(Sj ). Therefore, for πn defined as the coalition
S
structure containing coalitions { tr=1 Sbr ∪ T ∪ {n}} and
{Sj }, for all j ∈ [k] \ B, the requirement is fulfilled.
There are many possible imputations that can be chosen
and we focus on some of them that have a good stability
property:
j=1
Definition 5. [2] CS-core is the set of imputations hπ, xi
such that for any S ⊆ N , x(S) ≥ v(S).
For any imputation in the CS-core, no subset of agents S can
outperform x(S). Therefore no subset of agents S has an
incentive to abandon its respective coalitions in π to create
a new coalition. Indeed, even if the agent of a coalition
share their new outcomes, then at least one of them will not
benefit from this situation.
It is well known that imputation hπ, xi belongs to the CScore only if π is the optimal coalition structure. For bipolar
MC-nets, optimal coalition structure π ∗ can be computed in
polynomial time. However it remains for us to compute x to
find imputation hπ ∗ , xi that belongs to the core. The following proposition shows that we can focus on the computation
of the core for each coalition of π ∗ .
Proposition 7. For game v representable by bipolar MCnet, if π ∗ = {S1 , . . . , Sk } is the optimal coalition structure then hπ ∗ , xi belongs to the CS-core for any x such that
x(Ti ) ≥ v(Ti ) holds for any i ∈ [k] and any Ti ⊆ Si .
Proof. We need to show that for any T ⊆ N , x(T ) ≥
v(T ) holds. For any i ∈ [k], let Ti be the subset of agents in T
belonging to Si i.e., Ti = T ∩ Si . We know by Lemma 2 (see
S
S
Appendix) that v( ij=1 Sj ∪ kj=i+1 Tj ) + v(Ti ) ≥ v(Si ) +
Si−1
Sk
v( j=1 Sj ∪ j=i Tj ) holds, for any i ∈ [k]. By summing
S
P
up those inequalities, we obtain v( kj=1 Sj ) + kj=1 v(Tj ) ≥
Pk
Sk
j=1 v(Sj ) + v( j=1 Tj ). Furthermore, the optimality of
P
S
π ∗ implies that kj=1 v(Sj ) ≥ v( kj=1 Sj ). By summing up
P
S
these two inequalities, we obtain kj=1 v(Tj ) ≥ v( kj=1 Tj ) =
Pk
v(T ). Finally, this last inequality implies x(T ) = j=1 x(Tj ) ≥
Pk
j=1 v(Tj ) ≥ v(T ).
According to Proposition 7, we can assume without loss
of generality that π ∗ = {N }, because otherwise the CScore can be computed as the concatenation of imputations
belonging to the cores of each coalition of π ∗ . The following
proposition provides a formula to compute one particular
imputation belonging to the CS-core:
Proposition 8 also implies that the CS-core is non empty
for any game represented by a bipolar MC-net. We must
show how vector x can be computed in polynomial time.
The values of x are computed iteratively, starting from x1
and ending with xn . To compute xi , given the values of
x1 , . . . , xi−1 , we define set function qi : 2[i−1] → R such that
qi (T ) = v(T ∪ {i}) − x(T ), for any T ⊆ [i − 1]. xi should
clearly equal maxT ⊆[i−1] qi (T ). The following proposition
shows that these set functions are supermodular, which justifies resorting to a polynomial time algorithm (e.g., [26]) to
compute vector x.
Proposition 9. For any i ∈ N , qi is supermodular.
Proof. For any A, B ⊆ [i − 1], such that B ⊆ A, and
for any j ∈ [i − 1] \ A, we have qi (A) + qi (B ∪ {j}) =
v(A∪{i})−x(A)+v(B∪{j}∪{i})−x(B∪{j}) ≤ v(B∪{i})−
x(B) + v(A ∪ {j} ∪ {i}) − x(A ∪ {j}) = qi (B) + qi (A ∪ {j}),
where the inequality is due to Lemma 2 (see Appendix).
6.
CONCLUSION
This paper provides a partial mapping of the hard and
easy cases for the CSG problem for games represented by
basic MC-nets with positive weights. Some parameters of
MC-net representation, such as the number of red edges in
the CSG-graph, may be a source of complexity in the CSG
problem. The paper also provides an algorithm that solves
the CSG problem for a broader class of MC-nets than previous work [25], and studies the polynomial time cases that
arise from this new type of optimization. Finally it introduces a subclass of MC-nets, called bipolar MC-nets, which
represent games with a non-empty CS-core. The CSG problem as well as the determination of an imputation of the
CS-core are problems solvable in polynomial time.
This paper raises several open questions. The CSG problem’s complexity remains open for basic MC-nets with positive weights when the number of red edges in the CSG-graph
is restricted to one7 or two. Another question which de-
Proposition 8. For any game v representable
P by a bipolar MC-net, if xi = maxT ⊆[i−1] {v(T ∪ {i}) − j∈T xj } for
any i ∈ N , then hπ ∗ , xi belongs to the CS-core.
Proof. Let S be a subset of N , and let i be the highest
index
in S. By construction, we know that xi ≥ v(S) −
P
j∈S\{i} xj . This implies that x(S) ≥ v(S) holds for any
S ⊆ N . It remains for us to check that hπ ∗ , xi is an imputation, i.e. x(N ) = v(N ) holds. We show that there exists
partition πn of N such that x(N ) ≤ v(πn ). Then the optimality of coalition structure {N } implies x(N ) ≤ v(πn ) ≤
v(N ) ≤ x(N ), and the result follows.
We prove the existence of πn by induction on the number of agents. For one agent this statement holds trivially. Assume now that the statement is true for n − 1
agents, and let us show that it is also true for n agents.
By definition of x, we know that
P there exists T ⊆ [n − 1]
such that xn = v(T ∪ {n}) − j∈T xj , which implies that
7
Whenever the basic MC-net does not contain any rule without positive literals, this case is solvable in polynomial time.
314
Figure 2: Example of π(A) and π(B)
Figure 3: example after adding j to A and B
√
serves investigation is the possibility of designing an o( k)approximation algorithm, where k is the number of red edges
in the CSG-graph. Finally, we know that characteristic functions are fully expressive and that the CSG problem can be
solved in polynomial time in the size of this representation.
However this representation is not compact. It would be
interesting to design a fully expressive language that is as
compact as possible and where the CSG problem can be
solved in polynomial time. Bipolar MC-nets may be a good
starting point to explore this direction.
(6) (the latter exceeding (5)):
[
X
[
X
v̂(
S) −
v̂(S) − v̂(
S) +
v̂(S)
S∈D
Appendix
i∈P̄
S∩Ni =∅
α(S)∩Ni 6=∅
Lemma 1. Let j ∈ P and A, B ⊆ P \{j} such that B ⊆
A. Then the following inequality holds:
Proof. Let D ⊆ π(A) be the set of coalitions such that
∀S ∈ D, S ∩ Pj 6= ∅ and ∀S 0 ∈ π(A)\D, S 0 ∩ Pj = ∅. For any
S 0 ∈ π(A)\D we have S 0 ∈ π(A ∪ {j}), therefore:
[
X
v̂(π(A ∪ {j})) − v̂(π(A)) = v̂(
S) −
v̂(S)
(4)
S∈D
Figures 2 and 3 can help to figure out where come from
the r.h.s. of (4). The coalition belonging to π(A)\D does
not change by adding rule j to the set of rules that has to
hold (left part of Figs. 2 and 3). The only coalition
left for
S
π(A ∪ {j}) is the one containing Pj which is S∈D S (right
part of Fig. 3). Let D0 ⊆ π(B) be the set of coalitions such
that ∀S ∈ D0 , S ∩ Pj 6= ∅ and ∀S 0 ∈ π(B)\D0 , S 0 ∩ Pj = ∅.
For any S 0 ∈ π(B)\D0 we have S 0 ∈ π(B ∪ {j}), therefore:
[
X
v̂(π(B ∪ {j})) − v̂(π(B)) = v̂(
S) −
v̂(S)
(5)
S∈D 0
S∈D
S∈D 00
i∈P̄
S
( S∈D00 S)∩Ni =∅
S
( S∈D S)∩Ni 6=∅
S∈D
S∈D 00
S∈D 00
Finally, combining this inequality with (6), and using (4)
and (5), we obtain (3) that concludes the proof.
Lemma 2. For any bipolar game ∀A, B ⊆ N , A ∩ B 6=
∅ ⇒ v(A) + v(B) ≤ v(A ∩ B) + v(A ∪ B)
Proof. ∀A, B ⊆ N such that A ∩ B 6= ∅ we have:
X
X
X
X
vi +
vi ≤
vi +
vi
S∈D 0
The reasoning that justifies the r.h.s. of (5) is the same as
for (4) (in that case we need to isolate in Figs. 2 and 3
the elements of D0 instead of D). Select now D00 ⊆ D0 such
that ∀S ∈ D, |{S 00 |S 00 ∈ D00 , S 00 ⊆ S}| = 1 (there are various
possible sets D00 and the proof holds for any of them). Fig.
2 illustrates one possible
way of choosing D00 within D0 .
00
0 S
We have D ⊆ D , S∈D00 S 6= ∅ and ∀S ∈ D0 , v̂(S) ≥ 0,
therefore:
[
X
[
X
v̂(
S) −
v̂(S) ≤ v̂(
S) −
v̂(S)
(6)
S∈D 0
(7)
S∈D 00
This quantity
is non-negative Sbecause for any i ∈ P̄ such
S
that ( S∈D00 S)∩Ni = ∅ and ( S∈D S)∩Ni 6= ∅ there exists
at least one S ∈ D00 such that S ∩ Ni = ∅ and α(S) ∩ Ni 6= ∅.
00
Let us prove this by contradiction. Assume
S that 6 ∃S ∈ D
such that S ∩Ni = ∅ and α(S)∩Ni 6= ∅. ( S∈D00 S)∩Ni = ∅
implies that S ∩ Ni = ∅ holds ∀S ∈ D00 . So α(S) ∩ Ni = ∅
necessarilySholds for any SS∈ D00 , but this yields
a contradicS
tion since S∈D00 α(S) = S∈D S implies S∈D S ∩ Ni 6= ∅.
Thus, by combining (7) and (8) we obtain:
[
X
[
X
v̂(
S) −
v̂(S) ≥ v̂(
S) −
v̂(S)
v̂(π(A ∪ {j})) − v̂(π(A)) ≥ v̂(π(B ∪ {j})) − v̂(π(B)) (3)
S∈D
S∈D 00
For any S ∈ D00 let α(S) ∈ D be the coalition of D such
that S ⊆ α(S). By the definition of D00 for any S ∈ D00
exists a unique α(S) ∈ D (this fact is illustrated in Fig. 2)
so by the definition of v̂, Eq. (7) reads:
X
X
X
vi −
vi
(8)
S∈D 00
S∈D 0
S∈D
i∈P
i∈P
i∈P
i∈P
Pi ⊆A
Pi ⊆B
Pi ⊆A∩B
Pi ⊆A∪B
∀i ∈ P, (Pi ⊆ A) ∧ (Pi ⊆ B) ⇒ Pi ⊆ A ∩ B and (Pi ⊆
A) ∨ (Pi ⊆ B) ⇒ Pi ⊆ A ∪ B. Furthermore we have v̂(A) +
v̂(B) ≤ v̂(A∩B)+ v̂(A∪B). ∀i ∈ P̄ , (P̄i ∩A = ∅)∧(P̄i ∩B =
∅) ⇒ Ni ∩ (A ∪ B) = ∅ and (Ni ∩ S = ∅) ∨ (Ni ∩ B =
∅) ⇒ Ni ∩ (A ∩ B) = ∅ (no negative vi is involved since
A ∩ B 6= ∅). By adding these two inequalities we obtain
v(A) + v(B) ≤ v(A ∪ B) + v(A ∩ B) which concludes the
proof.
S∈D 00
Consider now the difference between the r.h.s. of (4) and
315
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